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The given equation is:

\[
3(x + 9)^{\frac{3}{4}} = 24
\]

Let's solve this step by step.

### Step 1: Divide both sides of the equation by 3 to simplify:
\[
(x + 9)^{\frac{3}{4}} = \frac{24}{3}
\]
\[
(x + 9)^{\frac{3}{4}} = 8
\]

### Step 2: Raise both sides to the power of \(\frac{4}{3}\) to eliminate the exponent:
\[
(x + 9) = 8^{\frac{4}{3}}
\]

### Step 3: Calculate \(8^{\frac{4}{3}}\):
\[
8^{\frac{4}{3}} = (2^3)^{\frac{4}{3}} = 2^4 = 16
\]
So,
\[
x + 9 = 16
\]

### Step 4: Solve for \(x\):
\[
x = 16 - 9
\]
\[
x = 7
\]

Thus, the solution is \(x = 7\). The correct answer is **C. 7**.

Would you like further details on any part of this solution?

Here are some related questions to deepen your understanding:

1. How do you solve an equation with fractional exponents in general?
2. What is the process to simplify fractional exponents?
3. What if the exponent had been a different fraction, such as \(\frac{2}{3}\)?
4. Can you explain the relationship between radical expressions and fractional exponents?
5. What would happen if the equation had a negative exponent?

**Tip**: When dealing with fractional exponents, remember that the denominator indicates the root, and the numerator indicates the power.

What is the solution to the equation 3(x + 9)^(3/4) = 24?

Learn how to solve an algebraic equation with fractional exponents, like 3(x + 9)^(3/4) = 24. This problem covers key concepts of fractional exponents, exponent laws, and algebraic manipulation. Suitable for Grades 9-11 students.

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